A Mott insulator is the prototypical example of a correlated insulator. At temperatures below the Mott gap, charge fluctuations are frozen while local spin moments remain disordered and nearly decoupled. The absence of exponentially localized orbitals to form such local moments in a topological band leads naturally to the question: is a Mott insulator possible in a topological band? I will begin by discussing a few general arguments constraining the nature of a Mott state in a topological band. I will then introduce a model of concentrated charge in a topological flat band, motivated by twisted bilayer graphene, where the scale of charge concentration provides a natural small parameter. I will show that while this parameter is non-perturbative, it is possible to perform analytically controlled calculations. By calculating the entropy and spectral functions at finite temperature exactly to leading order in the small parameter, I will establish that the model has a novel "Mott semimetal" phase at intermediate temperature scales with the following properties: (i) the spin is disordered and the entropy is consistent with that of decoupled fluctuating moments, (ii) the spectral function is gapped everywhere except for a parametrically small region which features a gapless Dirac cone, (iii) the Dirac cone responds to perturbation in a manner that is incompatible with any weakly interacting or mean field state. The Mott semimetal can be understood as the disordered phase of nonlocal spin moments with parametrically small overlap despite having non-perturbative power-law tails due to topology. At the end of my talk, I will discuss a novel numerical approach to study correlations in topological bands based on the bootstrap method. I will show that this approach can capture many of the exotic phases of the interacting lowest Landau level including Laughlin states and composite Fermi liquid and discuss how it can be applied to other systems.